Given that most people with serious food allergies seem to be "born allergic", seems unlikely. I was born allergic to cow's milk (it's a specific allergy to the type of milk protein produced by cows; goat and sheep's milk are both fine). Both of my children had the same allergy, which was evident immediately that I weaned them. It seems to wane at puberty, although I still have a million environmental allergies. This time of year is very difficult, because so many places have evergreen trees/wreaths/boughs inside, to which I am desperately allergic. Spend a lot of time this time of year coughing and wheezing.
This kind of immunotherapy for food allergies makes sense, because we know that "allergy shots" work for environmental allergies, and it's the same principle of introducing a small amount of the allergen to build up tolerance to it. And it works. I was very allergic to cats and dogs as a child, yet, I've had them most of my life, and in my experience, when I get a new critter, I usually have to suffer a bit for a couple of months until I get desensitized to the "new" dander. But with the threat of anaphylaxis, you'd have to be super careful, and very experienced at when/how to use the epipen and/or rush to emerg.